Benford's law states that the probability distribution of the first digits of many items (e.g. populations and expenses) is not uniform, but has the probabilities shown in this table. Business expenses tend to follow Benford's Law, because there are generally more small expenses than large expenses. Perform a "Goodness of Fit" Chi-Squared hypothesis test (αα = 0.05) to see if these values are consistent with Benford's Law. If they are not consistent, it there might be embezzelment. Complete this table. X Observed Frequency (Counts) Benford's Law P(X) Expected Frequency (Counts) 1 31 .301 2 30 .176 3 17 .125 4 14 .097 5 3 .079 6 11 .067 7 4 .058 8 8 .051 9 3 .046 The sum of the observed frequencies is 121. H0H0: p1p1 = .301, p2p2 = .176, p3p3 = .125, p4p4 = .097, p5p5 = .079, p6p6 = .067, p7p7 = .058, p8p8 = .051, p9p9 = .046 H1H1: at least one is different Original Claim: H₀ H₁ Test Statistic = (Round to three decimal places.) P-value = (Round to four decimal places.) Decision: reject the null hypothesis fail to reject the null hypothesis Accept the null hypothesis Conclusion: There is sufficient evidence to warrant rejection of the claim that these values are consistent with Benford's Law. There is not sufficient evidence to warrant rejection of the claim that these values are consistent with Benford's Law. The sample data supports the claim that these values are consistent with Benford's Law. There is not sufficient data to support the claim that these values are consistent with Benford's Law.
Contingency Table
A contingency table can be defined as the visual representation of the relationship between two or more categorical variables that can be evaluated and registered. It is a categorical version of the scatterplot, which is used to investigate the linear relationship between two variables. A contingency table is indeed a type of frequency distribution table that displays two variables at the same time.
Binomial Distribution
Binomial is an algebraic expression of the sum or the difference of two terms. Before knowing about binomial distribution, we must know about the binomial theorem.
Benford's law states that the probability distribution of the first digits of many items (e.g. populations and expenses) is not uniform, but has the probabilities shown in this table.
Business expenses tend to follow Benford's Law, because there are generally more small expenses than large expenses.
Perform a "Goodness of Fit" Chi-Squared hypothesis test (αα = 0.05) to see if these values are consistent with Benford's Law.
If they are not consistent, it there might be embezzelment.
Complete this table.
| X | Observed Frequency (Counts) |
Benford's Law P(X) |
Expected Frequency (Counts) |
|---|---|---|---|
| 1 | 31 | .301 | |
| 2 | 30 | .176 | |
| 3 | 17 | .125 | |
| 4 | 14 | .097 | |
| 5 | 3 | .079 | |
| 6 | 11 | .067 | |
| 7 | 4 | .058 | |
| 8 | 8 | .051 | |
| 9 | 3 | .046 |
The sum of the observed frequencies is 121.
H0H0: p1p1 = .301, p2p2 = .176, p3p3 = .125, p4p4 = .097, p5p5 = .079, p6p6 = .067, p7p7 = .058, p8p8 = .051, p9p9 = .046
H1H1: at least one is different
Original Claim:
- H₀
- H₁
Test Statistic =
(Round to three decimal places.) P-value =
(Round to four decimal places.) Decision:
- reject the null hypothesis
- fail to reject the null hypothesis
- Accept the null hypothesis
Conclusion:
- There is sufficient evidence to warrant rejection of the claim that these values are consistent with Benford's Law.
- There is not sufficient evidence to warrant rejection of the claim that these values are consistent with Benford's Law.
- The sample data supports the claim that these values are consistent with Benford's Law.
- There is not sufficient data to support the claim that these values are consistent with Benford's Law.
Enter the critical value, along with the significance level and degrees of freedom χ2χ2 (αα,df) below the graph. (Graph is for illustration only. No need to shade.)
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